In this paper, we study how a budget-constrained bidder should learn to bid adaptively in repeated first-price auctions to maximize cumulative payoff. This problem arises from the recent industry-wide shift from second-price auctions to first-price auctions in display advertising, which renders truthful bidding suboptimal. We propose a simple dual-gradient-descent-based bidding policy that maintains a dual variable for the budget constraint as the bidder consumes the budget. We analyze two settings based on the bidder's knowledge of future private values: (i) an uninformative setting where all distributional knowledge (potentially non-stationary) is entirely unknown, and (ii) an informative setting where a prediction of budget allocation is available in advance. We characterize the performance loss (regret) relative to an optimal policy with complete information. For uninformative setting, we show that the regret is ~O(sqrt(T)) plus a Wasserstein-based variation term capturing non-stationarity, which is order-optimal. In the informative setting, the variation term can be eliminated using predictions, yielding a regret of ~O(sqrt(T)) plus the prediction error. Furthermore, we go beyond the global budget constraint by introducing a refined benchmark based on a per-period budget allocation plan, achieving exactly ~O(sqrt(T)) regret. We also establish robustness guarantees when the baseline policy deviates from the planned allocation, covering both ideal and adversarial deviations.

翻译：本文研究预算受限的竞标者在重复第一价格拍卖中如何自适应学习出价以最大化累积收益。该问题源于显示广告领域近期从第二价格拍卖向第一价格拍卖的行业转型，这使得真实报价策略不再最优。我们提出一种基于双梯度下降的简单出价策略，该策略在竞标者消耗预算时维护一个针对预算约束的对偶变量。我们基于竞标者对未来私有价值的了解程度分析两种设置：（i）无信息设置，即所有分布知识（可能非平稳）完全未知；（ii）有信息设置，即预先获得预算分配的预测。我们刻画了相对于拥有完全信息的最优策略的性能损失（遗憾）。对于无信息设置，我们证明遗憾为~O(sqrt(T))加上一个刻画非平稳性的Wasserstein变分项，且该界阶最优。在有信息设置中，利用预测可消除变分项，得到~O(sqrt(T))加预测误差的遗憾界。进一步，我们超越全局预算约束，引入基于每期预算分配方案的精细化基准，实现了精确的~O(sqrt(T))遗憾界。我们还建立了基线策略偏离计划分配时的鲁棒性保证，涵盖理想情形和对抗性偏离。